# Universal Properties I - Inspiring Examples

The general fact of the uniqueness of the universal arrows implies the uniqueness of the ... object, up to a unique isomorphism (who wants more?).

Many (perhaps even most!) constructions in abstract algebra involve an object (and morphism) that satisfies some defining "universal property." But often the mathematician unaccustomed with category theory is left to wonder:

- What exactly
*is*a universal property? - Does every object have a universal property?
- Can an object have more than one universal property?
- Why do we care about universal properties, anyway?

We'll soon see answers to all of these questions, but for now let's just remember the first motto of category theory: it's all about the arrows!

Rather than dive straight into the technical definitions, I think it's more illuminating to first review many specific situations in which a common construction is characterized by some property about maps.

# Examples in

You encounter many "basic" constructions on sets as soon as you start studying set theory. Here we list a few, along with the universal property they satisfy.

## Cartesian product

Suppose

The map

## Disjoint union

The disjoint union of two sets is exactly dual to the Cartesian product construction. Indeed, if

## Quotient by equivalence relations

Suppose

(This example can be made even more "categorical" if one is willing to reframe the definition of "equivalence relation" without reference to elements.)

## Equalizers

Have you ever wondered if there was a set-analogue of the notion of a kernel? Since sets do not have an element identified as "zero", there is no direct analogue of a kernel. However, there is something close, called the *equalizer*. Given a pair of set maps **equalizer** of

At the level of elements, the map

## Coequalizers

As will always be the case, there is a notion dual to equalizers, named ... (drumroll) ... *coequalizers*. I will simply leave a diagram here with no explanation. Can you fill in the details?

## Pushouts/Pushforwards

Suppose

This information is universal among all such pairs of set maps from

## Pullbacks

Exactly dual to the notion of a pushout is that of a **pullback**. The relevant diagram is shown below. Can you fill in the details?

The pullback object is sometimes also called a **fibered product** or a **product of and over **.

# Examples in

## Quotient groups

The quotient group construction is probably the first time most of us officially encounter the phrase "universal property." Let's recall the details. Suppose

(You might recall that we actually have

You might notice something a little different about this example compared to all the others listed here. In this example, the universal property involves information *not* encoded in the diagram, namely that the arrows have been restricted (to only morphisms with a certain property). This suggests that we should ultimately look for a characterization of "universal property" that goes behind representing diagrams.

Now the quotient group is usually described as a group whose elements are cosets. However, once the cosets are used to prove this one "universal" property of

# Examples in

## Direct sum

Suppose **direct sum** of

Look at this diagram and compare with the diagram (in

At least as far as category theory is concerned ...

## Suggested next note

Universal Properties II - Commutative diagrams, cones and limits